(State-feedback controller):
The control law (REF ) has been widely studied for tracking control designs for mechanical systems in Euclidean space, particularly for robot manipulators. The proposed 4-DOF Lagrangian approach allows to bridge up the gap between control designs for mechanical systems in Euclidean space and control designs for the attitude of a rigid body in the quaternion group, and to enable using the whole quaternion in attitude control designs, in contrast to control designs that use only the vector part of the quaternion [1]}, [2]}, avoiding the topological obstruction discussed in [3]}, [4]} and singularities in controller development using any 3-DOF Lagrangian dynamics to describe the attitude of a rigid body [5]}, [6]}, [7]}, [1]}, [9]}. The global exponential convergence of the tracking error by the continuous controller (REF ) is achieved on the Euclidean space \(\mathbb {R}^4\)  by designing the control law that renders the origin \((e,\ \dot{e})=0\)  globally exponentially stable. Notice that \(q(t), \ q_{d}(t)\in \mathcal {S}^{3}\)  for all \(t\ge 0\) , so does the quaternion error \(\varepsilon (t)\)  defined in (REF ). The convergence to the origin of \(e(t)\)  implies the convergence of the quaternion error \(\varepsilon (t)\)  to the quaternion identity in the quaternion group \(\mathcal {S}^{3}\) .
